Imaging a Black Hole using only Closure Quantities

In this tutorial, we will create a preliminary reconstruction of the 2017 M87 data on April 6 using closure-only imaging. This tutorial is a general introduction to closure-only imaging in Comrade. For an introduction to simultaneous image and instrument modeling, see Stokes I Simultaneous Image and Instrument Modeling

Introduction to Closure Imaging

The EHT is the highest-resolution telescope ever created. Its resolution is equivalent to roughly tracking a hockey puck on the moon when viewing it from the earth. However, the EHT is also a unique interferometer. For one, the data it produces is incredibly sparse. The array is formed from only eight geographic locations around the planet, each with its unique telescope. Additionally, the EHT observes at a much higher frequency than typical interferometers. As a result, it is often difficult to directly provide calibrated data since the source model can be complicated. This implies there can be large instrumental effects often called gains that can corrupt our signal. One way to deal with this is to fit quantities that are independent of gains. These are often called closure quantities. The types of closure quantities are briefly described in Introduction to the VLBI Imaging Problem.

In this tutorial, we will do closure-only modeling of M87 to produce preliminary images of M87.

To get started, we will load Comrade

using Comrade




using Pyehtim
  Activating project at `~/work/Comrade.jl/Comrade.jl/examples`

For reproducibility we use a stable random number genreator

using StableRNGs
rng = StableRNG(123)
StableRNGs.LehmerRNG(state=0x000000000000000000000000000000f7)

Load the Data

To download the data visit https://doi.org/10.25739/g85n-f134 To load the eht-imaging obsdata object we do:

obs = ehtim.obsdata.load_uvfits(joinpath(dirname(pathof(Comrade)), "..", "examples", "SR1_M87_2017_096_lo_hops_netcal_StokesI.uvfits"))
Python: <ehtim.obsdata.Obsdata object at 0x7fc68fb568f0>

Now we do some minor preprocessing:

  • Scan average the data since the data have been preprocessed so that the gain phases are coherent.
  • Add 1% systematic noise to deal with calibration issues that cause 1% non-closing errors.
obs = scan_average(obs).add_fractional_noise(0.015)
Python: <ehtim.obsdata.Obsdata object at 0x7fc68fb56e60>

Now, we extract our closure quantities from the EHT data set.

dlcamp, dcphase  = extract_table(obs, LogClosureAmplitudes(;snrcut=3), ClosurePhases(;snrcut=3))
(EHTObservation{Float64,Comrade.EHTLogClosureAmplitudeDatum{Float64}, ...}
  source: M87
  mjd: 57849
  frequency: 2.27070703125e11
  bandwidth: 1.856e9
  stations: [:AA, :AP, :AZ, :JC, :LM, :PV, :SM]
  nsamples: 128
, EHTObservation{Float64,Comrade.EHTClosurePhaseDatum{Float64}, ...}
  source: M87
  mjd: 57849
  frequency: 2.27070703125e11
  bandwidth: 1.856e9
  stations: [:AA, :AP, :AZ, :JC, :LM, :PV, :SM]
  nsamples: 152
)

Build the Model/Posterior

For our model, we will be using an image model that consists of a raster of point sources, convolved with some pulse or kernel to make a ContinuousImage object with it Comrade's. generic image model.

function sky(θ, metadata)
    (;fg, c, σimg) = θ
    (;K, meanpr, grid, cache) = metadata
    # Construct the image model we fix the flux to 0.6 Jy in this case
    cp = meanpr .+ σimg.*c.params
    rast = ((1-fg))*K(to_simplex(CenteredLR(), cp))
    img = IntensityMap(rast, grid)
    m = ContinuousImage(img, cache)
    # Add a large-scale gaussian to deal with the over-resolved mas flux
    g = modify(Gaussian(), Stretch(μas2rad(250.0), μas2rad(250.0)), Renormalize(fg))
    return m + g
end
sky (generic function with 1 method)

Now, let's set up our image model. The EHT's nominal resolution is 20-25 μas. Additionally, the EHT is not very sensitive to a larger field of views; typically, 60-80 μas is enough to describe the compact flux of M87. Given this, we only need to use a small number of pixels to describe our image.

npix = 32
fovxy = μas2rad(150.0)
7.27220521664304e-10

Now, we can feed in the array information to form the cache

grid = imagepixels(fovxy, fovxy, npix, npix)
buffer = IntensityMap(zeros(npix,npix), grid)
cache = create_cache(NFFTAlg(dlcamp), buffer, BSplinePulse{3}())
VLBISkyModels.NUFTCache{VLBISkyModels.ObservedNUFT{NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}, Matrix{Float64}}, NFFT.NFFTPlan{Float64, 2, 1}, Vector{ComplexF64}, BSplinePulse{3}, KeyedArray{Float64, 2, NamedDimsArray{(:X, :Y), Float64, 2, Matrix{Float64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}}}(VLBISkyModels.ObservedNUFT{NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}, Matrix{Float64}}(NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}(1, 4, 2.0, :kaiser_bessel, AbstractNFFTs.POLYNOMIAL, true, false, true, 0x00000000), [-4.405690154666661e9 787577.6145833326 … -5.999801315555549e9 -15551.297851562484; -4.523017159111106e9 -1.6838098888888871e6 … 3.059254300444441e9 118294.64453124987]), NFFTPlan with 274 sampling points for an input array of size(32, 32) and an output array of size(274,) with dims 1:2, ComplexF64[0.7014670924816703 - 0.519511453845776im, 0.9999999862106403 - 6.398624051637031e-5im, 0.6947985314160843 - 0.5233374146500747im, 0.9328845732113544 - 0.16336912604935844im, 0.9329421134238738 + 0.16331744240563614im, 0.6948519110447875 + 0.5233083088067162im, 0.7925669922789302 + 0.40478845348863063im, 0.9999999864389528 - 6.519719206334107e-5im, 0.9340555032742773 - 0.1520910238460368im, 0.6896791530194316 - 0.5261728587493705im  …  0.8376666586657548 + 0.1784956664688869im, 0.9196690775195503 + 0.09173305645579018im, 0.9135584686897926 + 0.29199716871629755im, 0.9135610039654034 + 0.2919905620278203im, 0.8899638273254515 + 0.28719490128071723im, 0.9568100109264257 + 0.09814170760184636im, 0.8899622135777404 + 0.287201492222944im, 0.9925668796456906 + 0.0027765725149323534im, 0.8376650833595861 - 0.17848899851834446im, 0.9999999999247222 + 7.335331210317892e-6im], BSplinePulse{3}(), [0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0])

Now we need to specify our image prior. For this work we will use a Gaussian Markov Random field prior

using VLBIImagePriors, Distributions, DistributionsAD

Since we are using a Gaussian Markov random field prior we need to first specify our mean image. For this work we will use a symmetric Gaussian with a FWHM of 50 μas

fwhmfac = 2*sqrt(2*log(2))
mpr = modify(Gaussian(), Stretch(μas2rad(50.0)./fwhmfac))
imgpr = intensitymap(mpr, grid)
2-dimensional KeyedArray(NamedDimsArray(...)) with keys:
↓   X ∈ 32-element LinRange{Float64,...}Y ∈ 32-element LinRange{Float64,...}
And data, 32×32 NamedDimsArray(::Matrix{Float64}, (:X, :Y)):
                 (-3.52247e-10)(3.29522e-10)  (3.52247e-10)
 (-3.52247e-10)     6.37537e-8        1.32434e-7     6.37537e-8
 (-3.29522e-10)     1.32434e-7        2.751e-7       1.32434e-7
 (-3.06796e-10)     2.62014e-7        5.44273e-7     2.62014e-7
 (-2.84071e-10)     4.93724e-7        1.0256e-6      4.93724e-7
 (-2.61345e-10)     8.86092e-7   …    1.84065e-6     8.86092e-7
 (-2.38619e-10)     1.51463e-6        3.14629e-6     1.51463e-6
    ⋮                            ⋱    ⋮            
  (2.15894e-10)     2.46586e-6        5.12225e-6     2.46586e-6
  (2.38619e-10)     1.51463e-6        3.14629e-6     1.51463e-6
  (2.61345e-10)     8.86092e-7   …    1.84065e-6     8.86092e-7
  (2.84071e-10)     4.93724e-7        1.0256e-6      4.93724e-7
  (3.06796e-10)     2.62014e-7        5.44273e-7     2.62014e-7
  (3.29522e-10)     1.32434e-7        2.751e-7       1.32434e-7
  (3.52247e-10)     6.37537e-8        1.32434e-7     6.37537e-8

Now since we are actually modeling our image on the simplex we need to ensure that our mean image has unit flux

imgpr ./= flux(imgpr)

meanpr = to_real(CenteredLR(), Comrade.baseimage(imgpr))
metadata = (;meanpr,K=CenterImage(imgpr), grid, cache)
(meanpr = [-7.55422122563378 -6.823167558636962 … -6.823167558636962 -7.55422122563378; -6.823167558636962 -6.092113891640146 … -6.092113891640146 -6.823167558636962; … ; -6.823167558636962 -6.092113891640146 … -6.092113891640146 -6.823167558636962; -7.55422122563378 -6.823167558636962 … -6.823167558636962 -7.55422122563378], K = VLBIImagePriors.CenterImage{Matrix{Float64}, Tuple{Int64, Int64}}([0.9944957386363662 -0.005326704545454548 … 0.005326704545454572 0.005504261363636381; -0.005326704545454548 0.9948393969941349 … 0.005160603005865089 0.005326704545454565; … ; 0.005326704545454572 0.005160603005865089 … 0.9948393969941348 -0.005326704545454546; 0.005504261363636381 0.005326704545454565 … -0.005326704545454546 0.9944957386363638], (32, 32)), grid = GriddedKeys{(:X, :Y)}
	X: LinRange{Float64}(-3.5224744018114725e-10, 3.5224744018114725e-10, 32)
	Y: LinRange{Float64}(-3.5224744018114725e-10, 3.5224744018114725e-10, 32)
, cache = VLBISkyModels.NUFTCache{VLBISkyModels.ObservedNUFT{NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}, Matrix{Float64}}, NFFT.NFFTPlan{Float64, 2, 1}, Vector{ComplexF64}, BSplinePulse{3}, KeyedArray{Float64, 2, NamedDimsArray{(:X, :Y), Float64, 2, Matrix{Float64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}}}(VLBISkyModels.ObservedNUFT{NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}, Matrix{Float64}}(NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}(1, 4, 2.0, :kaiser_bessel, AbstractNFFTs.POLYNOMIAL, true, false, true, 0x00000000), [-4.405690154666661e9 787577.6145833326 … -5.999801315555549e9 -15551.297851562484; -4.523017159111106e9 -1.6838098888888871e6 … 3.059254300444441e9 118294.64453124987]), NFFTPlan with 274 sampling points for an input array of size(32, 32) and an output array of size(274,) with dims 1:2, ComplexF64[0.7014670924816703 - 0.519511453845776im, 0.9999999862106403 - 6.398624051637031e-5im, 0.6947985314160843 - 0.5233374146500747im, 0.9328845732113544 - 0.16336912604935844im, 0.9329421134238738 + 0.16331744240563614im, 0.6948519110447875 + 0.5233083088067162im, 0.7925669922789302 + 0.40478845348863063im, 0.9999999864389528 - 6.519719206334107e-5im, 0.9340555032742773 - 0.1520910238460368im, 0.6896791530194316 - 0.5261728587493705im  …  0.8376666586657548 + 0.1784956664688869im, 0.9196690775195503 + 0.09173305645579018im, 0.9135584686897926 + 0.29199716871629755im, 0.9135610039654034 + 0.2919905620278203im, 0.8899638273254515 + 0.28719490128071723im, 0.9568100109264257 + 0.09814170760184636im, 0.8899622135777404 + 0.287201492222944im, 0.9925668796456906 + 0.0027765725149323534im, 0.8376650833595861 - 0.17848899851834446im, 0.9999999999247222 + 7.335331210317892e-6im], BSplinePulse{3}(), [0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]))

In addition we want a reasonable guess for what the resolution of our image should be. For radio astronomy this is given by roughly the longest baseline in the image. To put this into pixel space we then divide by the pixel size.

beam = beamsize(dlcamp)
rat = (beam/(step(grid.X)))
5.326336637737519

To make the Gaussian Markov random field efficient we first precompute a bunch of quantities that allow us to scale things linearly with the number of image pixels. This drastically improves the usual N^3 scaling you get from usual Gaussian Processes.

crcache = MarkovRandomFieldCache(meanpr)
VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}(sparse([1, 2, 32, 33, 993, 1, 2, 3, 34, 994  …  31, 991, 1022, 1023, 1024, 32, 992, 993, 1023, 1024], [1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27, 27, 27, 28, 28, 28, 28, 28, 29, 29, 29, 29, 29, 30, 30, 30, 30, 30, 31, 31, 31, 31, 31, 32, 32, 32, 32, 32, 33, 33, 33, 33, 33, 34, 34, 34, 34, 34, 35, 35, 35, 35, 35, 36, 36, 36, 36, 36, 37, 37, 37, 37, 37, 38, 38, 38, 38, 38, 39, 39, 39, 39, 39, 40, 40, 40, 40, 40, 41, 41, 41, 41, 41, 42, 42, 42, 42, 42, 43, 43, 43, 43, 43, 44, 44, 44, 44, 44, 45, 45, 45, 45, 45, 46, 46, 46, 46, 46, 47, 47, 47, 47, 47, 48, 48, 48, 48, 48, 49, 49, 49, 49, 49, 50, 50, 50, 50, 50, 51, 51, 51, 51, 51, 52, 52, 52, 52, 52, 53, 53, 53, 53, 53, 54, 54, 54, 54, 54, 55, 55, 55, 55, 55, 56, 56, 56, 56, 56, 57, 57, 57, 57, 57, 58, 58, 58, 58, 58, 59, 59, 59, 59, 59, 60, 60, 60, 60, 60, 61, 61, 61, 61, 61, 62, 62, 62, 62, 62, 63, 63, 63, 63, 63, 64, 64, 64, 64, 64, 65, 65, 65, 65, 65, 66, 66, 66, 66, 66, 67, 67, 67, 67, 67, 68, 68, 68, 68, 68, 69, 69, 69, 69, 69, 70, 70, 70, 70, 70, 71, 71, 71, 71, 71, 72, 72, 72, 72, 72, 73, 73, 73, 73, 73, 74, 74, 74, 74, 74, 75, 75, 75, 75, 75, 76, 76, 76, 76, 76, 77, 77, 77, 77, 77, 78, 78, 78, 78, 78, 79, 79, 79, 79, 79, 80, 80, 80, 80, 80, 81, 81, 81, 81, 81, 82, 82, 82, 82, 82, 83, 83, 83, 83, 83, 84, 84, 84, 84, 84, 85, 85, 85, 85, 85, 86, 86, 86, 86, 86, 87, 87, 87, 87, 87, 88, 88, 88, 88, 88, 89, 89, 89, 89, 89, 90, 90, 90, 90, 90, 91, 91, 91, 91, 91, 92, 92, 92, 92, 92, 93, 93, 93, 93, 93, 94, 94, 94, 94, 94, 95, 95, 95, 95, 95, 96, 96, 96, 96, 96, 97, 97, 97, 97, 97, 98, 98, 98, 98, 98, 99, 99, 99, 99, 99, 100, 100, 100, 100, 100, 101, 101, 101, 101, 101, 102, 102, 102, 102, 102, 103, 103, 103, 103, 103, 104, 104, 104, 104, 104, 105, 105, 105, 105, 105, 106, 106, 106, 106, 106, 107, 107, 107, 107, 107, 108, 108, 108, 108, 108, 109, 109, 109, 109, 109, 110, 110, 110, 110, 110, 111, 111, 111, 111, 111, 112, 112, 112, 112, 112, 113, 113, 113, 113, 113, 114, 114, 114, 114, 114, 115, 115, 115, 115, 115, 116, 116, 116, 116, 116, 117, 117, 117, 117, 117, 118, 118, 118, 118, 118, 119, 119, 119, 119, 119, 120, 120, 120, 120, 120, 121, 121, 121, 121, 121, 122, 122, 122, 122, 122, 123, 123, 123, 123, 123, 124, 124, 124, 124, 124, 125, 125, 125, 125, 125, 126, 126, 126, 126, 126, 127, 127, 127, 127, 127, 128, 128, 128, 128, 128, 129, 129, 129, 129, 129, 130, 130, 130, 130, 130, 131, 131, 131, 131, 131, 132, 132, 132, 132, 132, 133, 133, 133, 133, 133, 134, 134, 134, 134, 134, 135, 135, 135, 135, 135, 136, 136, 136, 136, 136, 137, 137, 137, 137, 137, 138, 138, 138, 138, 138, 139, 139, 139, 139, 139, 140, 140, 140, 140, 140, 141, 141, 141, 141, 141, 142, 142, 142, 142, 142, 143, 143, 143, 143, 143, 144, 144, 144, 144, 144, 145, 145, 145, 145, 145, 146, 146, 146, 146, 146, 147, 147, 147, 147, 147, 148, 148, 148, 148, 148, 149, 149, 149, 149, 149, 150, 150, 150, 150, 150, 151, 151, 151, 151, 151, 152, 152, 152, 152, 152, 153, 153, 153, 153, 153, 154, 154, 154, 154, 154, 155, 155, 155, 155, 155, 156, 156, 156, 156, 156, 157, 157, 157, 157, 157, 158, 158, 158, 158, 158, 159, 159, 159, 159, 159, 160, 160, 160, 160, 160, 161, 161, 161, 161, 161, 162, 162, 162, 162, 162, 163, 163, 163, 163, 163, 164, 164, 164, 164, 164, 165, 165, 165, 165, 165, 166, 166, 166, 166, 166, 167, 167, 167, 167, 167, 168, 168, 168, 168, 168, 169, 169, 169, 169, 169, 170, 170, 170, 170, 170, 171, 171, 171, 171, 171, 172, 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892, 892, 892, 892, 893, 893, 893, 893, 893, 894, 894, 894, 894, 894, 895, 895, 895, 895, 895, 896, 896, 896, 896, 896, 897, 897, 897, 897, 897, 898, 898, 898, 898, 898, 899, 899, 899, 899, 899, 900, 900, 900, 900, 900, 901, 901, 901, 901, 901, 902, 902, 902, 902, 902, 903, 903, 903, 903, 903, 904, 904, 904, 904, 904, 905, 905, 905, 905, 905, 906, 906, 906, 906, 906, 907, 907, 907, 907, 907, 908, 908, 908, 908, 908, 909, 909, 909, 909, 909, 910, 910, 910, 910, 910, 911, 911, 911, 911, 911, 912, 912, 912, 912, 912, 913, 913, 913, 913, 913, 914, 914, 914, 914, 914, 915, 915, 915, 915, 915, 916, 916, 916, 916, 916, 917, 917, 917, 917, 917, 918, 918, 918, 918, 918, 919, 919, 919, 919, 919, 920, 920, 920, 920, 920, 921, 921, 921, 921, 921, 922, 922, 922, 922, 922, 923, 923, 923, 923, 923, 924, 924, 924, 924, 924, 925, 925, 925, 925, 925, 926, 926, 926, 926, 926, 927, 927, 927, 927, 927, 928, 928, 928, 928, 928, 929, 929, 929, 929, 929, 930, 930, 930, 930, 930, 931, 931, 931, 931, 931, 932, 932, 932, 932, 932, 933, 933, 933, 933, 933, 934, 934, 934, 934, 934, 935, 935, 935, 935, 935, 936, 936, 936, 936, 936, 937, 937, 937, 937, 937, 938, 938, 938, 938, 938, 939, 939, 939, 939, 939, 940, 940, 940, 940, 940, 941, 941, 941, 941, 941, 942, 942, 942, 942, 942, 943, 943, 943, 943, 943, 944, 944, 944, 944, 944, 945, 945, 945, 945, 945, 946, 946, 946, 946, 946, 947, 947, 947, 947, 947, 948, 948, 948, 948, 948, 949, 949, 949, 949, 949, 950, 950, 950, 950, 950, 951, 951, 951, 951, 951, 952, 952, 952, 952, 952, 953, 953, 953, 953, 953, 954, 954, 954, 954, 954, 955, 955, 955, 955, 955, 956, 956, 956, 956, 956, 957, 957, 957, 957, 957, 958, 958, 958, 958, 958, 959, 959, 959, 959, 959, 960, 960, 960, 960, 960, 961, 961, 961, 961, 961, 962, 962, 962, 962, 962, 963, 963, 963, 963, 963, 964, 964, 964, 964, 964, 965, 965, 965, 965, 965, 966, 966, 966, 966, 966, 967, 967, 967, 967, 967, 968, 968, 968, 968, 968, 969, 969, 969, 969, 969, 970, 970, 970, 970, 970, 971, 971, 971, 971, 971, 972, 972, 972, 972, 972, 973, 973, 973, 973, 973, 974, 974, 974, 974, 974, 975, 975, 975, 975, 975, 976, 976, 976, 976, 976, 977, 977, 977, 977, 977, 978, 978, 978, 978, 978, 979, 979, 979, 979, 979, 980, 980, 980, 980, 980, 981, 981, 981, 981, 981, 982, 982, 982, 982, 982, 983, 983, 983, 983, 983, 984, 984, 984, 984, 984, 985, 985, 985, 985, 985, 986, 986, 986, 986, 986, 987, 987, 987, 987, 987, 988, 988, 988, 988, 988, 989, 989, 989, 989, 989, 990, 990, 990, 990, 990, 991, 991, 991, 991, 991, 992, 992, 992, 992, 992, 993, 993, 993, 993, 993, 994, 994, 994, 994, 994, 995, 995, 995, 995, 995, 996, 996, 996, 996, 996, 997, 997, 997, 997, 997, 998, 998, 998, 998, 998, 999, 999, 999, 999, 999, 1000, 1000, 1000, 1000, 1000, 1001, 1001, 1001, 1001, 1001, 1002, 1002, 1002, 1002, 1002, 1003, 1003, 1003, 1003, 1003, 1004, 1004, 1004, 1004, 1004, 1005, 1005, 1005, 1005, 1005, 1006, 1006, 1006, 1006, 1006, 1007, 1007, 1007, 1007, 1007, 1008, 1008, 1008, 1008, 1008, 1009, 1009, 1009, 1009, 1009, 1010, 1010, 1010, 1010, 1010, 1011, 1011, 1011, 1011, 1011, 1012, 1012, 1012, 1012, 1012, 1013, 1013, 1013, 1013, 1013, 1014, 1014, 1014, 1014, 1014, 1015, 1015, 1015, 1015, 1015, 1016, 1016, 1016, 1016, 1016, 1017, 1017, 1017, 1017, 1017, 1018, 1018, 1018, 1018, 1018, 1019, 1019, 1019, 1019, 1019, 1020, 1020, 1020, 1020, 1020, 1021, 1021, 1021, 1021, 1021, 1022, 1022, 1022, 1022, 1022, 1023, 1023, 1023, 1023, 1023, 1024, 1024, 1024, 1024, 1024], [4.0, -1.0, -1.0, -1.0, -1.0, -1.0, 4.0, -1.0, -1.0, -1.0  …  -1.0, -1.0, -1.0, 4.0, -1.0, -1.0, -1.0, -1.0, -1.0, 4.0], 1024, 1024), [1.0 0.0 … 0.0 0.0; 0.0 1.0 … 0.0 0.0; … ; 0.0 0.0 … 1.0 0.0; 0.0 0.0 … 0.0 1.0], [0.0 0.03842943919353914 … 0.15224093497742697 0.03842943919353936; 0.03842943919353914 0.07685887838707828 … 0.1906703741709661 0.0768588783870785; … ; 0.15224093497742697 0.1906703741709661 … 0.30448186995485393 0.19067037417096633; 0.03842943919353914 0.07685887838707828 … 0.1906703741709661 0.0768588783870785])

One of the benefits of the Bayesian approach is that we can fit for the hyperparameters of our prior/regularizers unlike traditional RML appraoches. To construct this heirarchical prior we will first make a map that takes in our regularizer hyperparameters and returns the image prior given those hyperparameters.

fmap = let crcache=crcache
    x->GaussMarkovRandomField(x, 1.0, crcache)
end
#1 (generic function with 1 method)

Now we can finally form our image prior. For this we use a heirarchical prior where the correlation length is given by a inverse gamma prior to prevent overfitting. Gaussian Markov random fields are extremly flexible models. To prevent overfitting it is common to use priors that penalize complexity. Therefore, we want to use priors that enforce similarity to our mean image, and prefer smoothness.

cprior = HierarchicalPrior(fmap, InverseGamma(1.0, -log(0.01*rat)))

prior = NamedDist(c = cprior, σimg = truncated(Normal(0.0, 1.0); lower=0.01), fg=Uniform(0.0, 1.0))

lklhd = RadioLikelihood(sky, dlcamp, dcphase;
                        skymeta = metadata)
post = Posterior(lklhd, prior)
Posterior{RadioLikelihood{Comrade.ModelMetadata{typeof(Main.sky), NamedTuple{(:meanpr, :K, :grid, :cache), Tuple{Matrix{Float64}, VLBIImagePriors.CenterImage{Matrix{Float64}, Tuple{Int64, Int64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}, VLBISkyModels.NUFTCache{VLBISkyModels.ObservedNUFT{NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}, Matrix{Float64}}, NFFT.NFFTPlan{Float64, 2, 1}, Vector{ComplexF64}, BSplinePulse{3}, KeyedArray{Float64, 2, NamedDimsArray{(:X, :Y), Float64, 2, Matrix{Float64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}}}}}}, Nothing, Tuple{Comrade.EHTObservation{Float64, Comrade.EHTLogClosureAmplitudeDatum{Float64}, StructArrays.StructVector{Comrade.EHTLogClosureAmplitudeDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :U4, :V4, :T, :F, :quadrangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{NTuple{4, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}, Comrade.EHTObservation{Float64, Comrade.EHTClosurePhaseDatum{Float64}, StructArrays.StructVector{Comrade.EHTClosurePhaseDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :T, :F, :triangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}}, Tuple{Comrade.ConditionedLikelihood{Comrade.var"#34#35"{Float64, Base.Fix2{typeof(logclosure_amplitudes), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}, Comrade.ConditionedLikelihood{Comrade.var"#36#37"{Float64, Base.Fix2{typeof(closure_phases), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, NamedTuple{(:U, :V, :T, :F), NTuple{4, Vector{Float64}}}}, VLBIImagePriors.NamedDist{(:c, :σimg, :fg), Tuple{VLBIImagePriors.HierarchicalPrior{Main.var"#1#2"{VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}}, Distributions.InverseGamma{Float64}}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}, Distributions.Uniform{Float64}}}}(RadioLikelihood
	Number of data products: 2
, VLBIImagePriors.NamedDist{(:c, :σimg, :fg), Tuple{VLBIImagePriors.HierarchicalPrior{Main.var"#1#2"{VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}}, Distributions.InverseGamma{Float64}}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}, Distributions.Uniform{Float64}}}(
dists: (VLBIImagePriors.HierarchicalPrior{Main.var"#1#2"{VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}}, Distributions.InverseGamma{Float64}}(
priormap: #1
hyperprior: Distributions.InverseGamma{Float64}(
invd: Distributions.Gamma{Float64}(α=1.0, θ=0.3410052124381751)
θ: 2.932506494109095
)

)
, Truncated(Distributions.Normal{Float64}(μ=0.0, σ=1.0); lower=0.01), Distributions.Uniform{Float64}(a=0.0, b=1.0))
)
)

Reconstructing the Image

To sample from this posterior, it is convenient to first move from our constrained parameter space to an unconstrained one (i.e., the support of the transformed posterior is (-∞, ∞)). This is done using the asflat function.

tpost = asflat(post)
Comrade.TransformedPosterior{Posterior{RadioLikelihood{Comrade.ModelMetadata{typeof(Main.sky), NamedTuple{(:meanpr, :K, :grid, :cache), Tuple{Matrix{Float64}, VLBIImagePriors.CenterImage{Matrix{Float64}, Tuple{Int64, Int64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}, VLBISkyModels.NUFTCache{VLBISkyModels.ObservedNUFT{NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}, Matrix{Float64}}, NFFT.NFFTPlan{Float64, 2, 1}, Vector{ComplexF64}, BSplinePulse{3}, KeyedArray{Float64, 2, NamedDimsArray{(:X, :Y), Float64, 2, Matrix{Float64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}}}}}}, Nothing, Tuple{Comrade.EHTObservation{Float64, Comrade.EHTLogClosureAmplitudeDatum{Float64}, StructArrays.StructVector{Comrade.EHTLogClosureAmplitudeDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :U4, :V4, :T, :F, :quadrangle), 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Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, 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TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}, Comrade.ConditionedLikelihood{Comrade.var"#36#37"{Float64, Base.Fix2{typeof(closure_phases), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, NamedTuple{(:U, :V, :T, :F), NTuple{4, Vector{Float64}}}}, VLBIImagePriors.NamedDist{(:c, :σimg, :fg), Tuple{VLBIImagePriors.HierarchicalPrior{Main.var"#1#2"{VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}}, Distributions.InverseGamma{Float64}}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}, Distributions.Uniform{Float64}}}}, TransformVariables.TransformTuple{NamedTuple{(:c, :σimg, :fg), Tuple{TransformVariables.TransformTuple{NamedTuple{(:params, :hyperparams), Tuple{TransformVariables.ArrayTransformation{TransformVariables.Identity, 2}, TransformVariables.ShiftedExp{true, Float64}}}}, TransformVariables.ShiftedExp{true, Float64}, TransformVariables.ScaledShiftedLogistic{Float64}}}}}(Posterior{RadioLikelihood{Comrade.ModelMetadata{typeof(Main.sky), NamedTuple{(:meanpr, :K, :grid, :cache), Tuple{Matrix{Float64}, VLBIImagePriors.CenterImage{Matrix{Float64}, Tuple{Int64, Int64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}, VLBISkyModels.NUFTCache{VLBISkyModels.ObservedNUFT{NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}, Matrix{Float64}}, NFFT.NFFTPlan{Float64, 2, 1}, Vector{ComplexF64}, BSplinePulse{3}, KeyedArray{Float64, 2, NamedDimsArray{(:X, :Y), Float64, 2, Matrix{Float64}}, GriddedKeys{(:X, :Y), Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}, ComradeBase.NoHeader, Float64}}}}}}, Nothing, Tuple{Comrade.EHTObservation{Float64, Comrade.EHTLogClosureAmplitudeDatum{Float64}, StructArrays.StructVector{Comrade.EHTLogClosureAmplitudeDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :U4, :V4, :T, :F, :quadrangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{NTuple{4, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}, Comrade.EHTObservation{Float64, Comrade.EHTClosurePhaseDatum{Float64}, StructArrays.StructVector{Comrade.EHTClosurePhaseDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :T, :F, :triangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, 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VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}, Comrade.ConditionedLikelihood{Comrade.var"#36#37"{Float64, Base.Fix2{typeof(closure_phases), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, NamedTuple{(:U, :V, :T, :F), NTuple{4, Vector{Float64}}}}, VLBIImagePriors.NamedDist{(:c, :σimg, :fg), Tuple{VLBIImagePriors.HierarchicalPrior{Main.var"#1#2"{VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}}, Distributions.InverseGamma{Float64}}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}, Distributions.Uniform{Float64}}}}(RadioLikelihood
	Number of data products: 2
, VLBIImagePriors.NamedDist{(:c, :σimg, :fg), Tuple{VLBIImagePriors.HierarchicalPrior{Main.var"#1#2"{VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}}, Distributions.InverseGamma{Float64}}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}, Distributions.Uniform{Float64}}}(
dists: (VLBIImagePriors.HierarchicalPrior{Main.var"#1#2"{VLBIImagePriors.MarkovRandomFieldCache{SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}}}, Distributions.InverseGamma{Float64}}(
priormap: #1
hyperprior: Distributions.InverseGamma{Float64}(
invd: Distributions.Gamma{Float64}(α=1.0, θ=0.3410052124381751)
θ: 2.932506494109095
)

)
, Truncated(Distributions.Normal{Float64}(μ=0.0, σ=1.0); lower=0.01), Distributions.Uniform{Float64}(a=0.0, b=1.0))
)
), TransformVariables.TransformTuple{NamedTuple{(:c, :σimg, :fg), Tuple{TransformVariables.TransformTuple{NamedTuple{(:params, :hyperparams), Tuple{TransformVariables.ArrayTransformation{TransformVariables.Identity, 2}, TransformVariables.ShiftedExp{true, Float64}}}}, TransformVariables.ShiftedExp{true, Float64}, TransformVariables.ScaledShiftedLogistic{Float64}}}}((c = TransformVariables.TransformTuple{NamedTuple{(:params, :hyperparams), Tuple{TransformVariables.ArrayTransformation{TransformVariables.Identity, 2}, TransformVariables.ShiftedExp{true, Float64}}}}((params = TransformVariables.ArrayTransformation{TransformVariables.Identity, 2}(asℝ, (32, 32)), hyperparams = as(Real, 0.0, ∞)), 1025), σimg = as(Real, 0.01, ∞), fg = as(Real, 0.0, 1.0)), 1027))

We can now also find the dimension of our posterior or the number of parameters we will sample.

Warning

This can often be different from what you would expect. This is especially true when using angular variables, where we often artificially increase the dimension of the parameter space to make sampling easier.

ndim = dimension(tpost)
1027

Now we optimize using LBFGS

using ComradeOptimization
using OptimizationOptimJL
using Zygote
f = OptimizationFunction(tpost, Optimization.AutoZygote())
prob = Optimization.OptimizationProblem(f, prior_sample(rng, tpost), nothing)
sol = solve(prob, LBFGS(); maxiters=5_00);

Before we analyze our solution we first need to transform back to parameter space.

xopt = transform(tpost, sol)
(c = (params = [-0.8494620261149127 -0.6903432132227583 … -1.1460429104665082 -0.9962305690916786; -0.804566006281114 -0.634332548000261 … -1.283397291236575 -1.025321655049105; … ; -0.7968146139720274 -0.6670668747733154 … -0.9086999181453329 -0.8422587408717032; -0.8009085653610114 -0.6496714575767522 … -1.025180006630898 -0.9163586462477289], hyperparams = 19.018518327712204), σimg = 3.381626657616295, fg = 0.6883212825680473)

First we will evaluate our fit by plotting the residuals

using Plots
residual(skymodel(post, xopt), dlcamp, ylabel="Log Closure Amplitude Res.")
Example block output

and now closure phases

residual(skymodel(post, xopt), dcphase, ylabel="|Closure Phase Res.|")
Example block output

Now let's plot the MAP estimate.

import WGLMakie as CM
img = intensitymap(skymodel(post, xopt), μas2rad(150.0), μas2rad(150.0), 100, 100)
CM.image(img, axis=(xreversed=true, aspect=1, title="MAP Image"), colormap=:afmhot, figure=(;resolution=(400, 400),))

To sample from the posterior we will use HMC and more specifically the NUTS algorithm. For information about NUTS see Michael Betancourt's notes.

Note

For our metric we use a diagonal matrix due to easier tuning.

using ComradeAHMC
using Zygote
metric = DiagEuclideanMetric(ndim)
chain, stats = sample(post, AHMC(;metric, autodiff=Val(:Zygote)), 700; nadapts=500, init_params=xopt)
(NamedTuple{(:c, :σimg, :fg), Tuple{NamedTuple{(:params, :hyperparams), Tuple{Matrix{Float64}, Float64}}, Float64, Float64}}[(c = (params = [-0.9462351771077331 -0.6975258664214178 … -1.1730488151042615 -0.9449714870358811; -0.7816151663792513 -0.6407229074028492 … -1.3506955920070498 -0.9467244787760345; … ; -0.7489769331050818 -0.6441778166434293 … -0.8975879826019837 -0.9176535201985786; -0.7737310566097165 -0.6333205468635623 … -1.0087926598289623 -0.8730363780347268], hyperparams = 19.63619547827477), σimg = 3.494541487272257, fg = 0.6926728712080034), (c = (params = [-1.062602295274311 -0.18389402297487253 … -0.3386383289846316 -0.4155995520874266; -1.678745155800979 -0.35676851515820796 … -1.2988249531740057 -1.5039115585085114; … ; 0.7549584545763055 0.06353538431614432 … -0.5376905170741717 0.2823331683787087; -0.7422585871564134 -1.3905389295783637 … -0.8146345723832336 0.057053443176966354], hyperparams = 31.433891010036312), σimg = 2.1997348600378057, fg = 0.6309006376052073), (c = (params = [0.2812715016552185 -0.5539572695879331 … -1.2258487689604667 -0.8840335560587025; 0.62440144192357 -0.18635402122274133 … -0.4124736028475872 -0.08381871736477267; … ; -2.214924760467808 -1.678849110673396 … -0.5398643492488352 -1.9518154555890765; -0.23122911721742487 0.48181583301542635 … -1.0131695548664728 -0.7285131806649956], hyperparams = 38.04112327917421), σimg = 1.4149257130828912, fg = 0.5018565346983168), (c = (params = [-0.37006334607382757 0.07589001913729276 … -0.36951967036442224 0.06377385370615883; -0.11261735184379339 -0.20683181164891096 … -0.6918506812287294 0.4212916609314396; … ; -0.9740281328720336 0.2861797177809091 … -0.6457868599644515 -1.6931635949599457; -0.4037985422102937 -1.2552317623379436 … -0.9928537924629096 -1.8541859234816496], hyperparams = 12.615921829124828), σimg = 1.7768591698585163, fg = 0.5691240838167585), (c = (params = [-0.8405708525422453 -1.2740293752263836 … -0.9125766427169607 -0.742266787355985; -0.8629609548462537 -0.7118666738072563 … -0.16444408872810176 -0.4793999064911864; … ; -0.5834855577349618 -0.8460351944969285 … -0.8221980917053939 -0.27572318958181186; -1.1350996649248917 0.46996423572804774 … -1.0894287991143086 -0.0828841077700087], hyperparams = 106.59029468628931), σimg = 1.5546822227498114, fg = 0.3199936444422504), (c = (params = [-0.8435739616227437 -1.2860056089667373 … -0.9232481289760317 -0.7420965688212912; -0.8612886652929286 -0.7110236059811342 … -0.15735491135772664 -0.4809794847286285; … ; -0.5626689451200156 -0.85230966836672 … -0.8401004765716991 -0.27540922233789145; -1.156501036191873 0.4781241909889329 … -1.0794318058729646 -0.08125868032010827], hyperparams = 107.53394399079035), σimg = 1.5471324434266416, fg = 0.3192209947160163), (c = (params = [-0.7736903714655502 -0.8748262184852462 … -0.0709770905241182 -1.1228833487899028; -0.6261088011411556 -1.3915563512411357 … -0.8797026145362267 -0.609946605113679; … ; -1.011754712991703 0.14014947901134414 … -2.294940205429686 -1.8526673516607304; -0.026051770996812072 -1.6913963231035478 … -0.44377475811685596 -1.192011886084195], hyperparams = 57.059044873347894), σimg = 1.4026805079248506, fg = 0.4508014653102825), (c = (params = [-0.8801850710691077 -1.0331611080320786 … -0.5956009024204342 -0.6597392968190569; -0.7756684145529806 -1.1666980931429352 … -0.6331432599175869 -0.5769706972148662; … ; -1.2511218912540245 -0.09394595275457243 … -2.074272076620974 -1.9228192263793513; 0.18634957548526512 -1.212502794672401 … -1.08972399656909 -1.0125051746537141], hyperparams = 88.4613591717273), σimg = 1.813071740875319, fg = 0.5107236364206656), (c = (params = [-1.2365013215556486 -1.5995068431300956 … -1.0051498854898686 -1.067953697360107; -1.1681986670469586 -0.8109018209748344 … -0.8357215241123835 -1.062377955222338; … ; -0.3352996052956795 -0.23738056180094916 … -2.225144854757743 -1.4709545825903656; -1.7634835302865617 -0.7087851152151795 … -1.9099213006216447 -1.2609478153242661], hyperparams = 15.512515138059593), σimg = 1.8091654405959747, fg = 0.4852486974931797), (c = (params = [-1.2365013215556486 -1.5995068431300956 … -1.0051498854898686 -1.067953697360107; -1.1681986670469586 -0.8109018209748344 … -0.8357215241123835 -1.062377955222338; … ; -0.3352996052956795 -0.23738056180094916 … -2.225144854757743 -1.4709545825903656; -1.7634835302865617 -0.7087851152151795 … -1.9099213006216447 -1.2609478153242661], hyperparams = 15.512515138059593), σimg = 1.8091654405959747, fg = 0.4852486974931797)  …  (c = (params = [0.021335566742223533 0.17794721664477375 … 0.194311886638769 -0.3157697757338721; -0.836220523343804 -0.07559949551382433 … -0.8886612757597931 -0.06400137235845674; … ; 0.3773842578887056 0.6288240580057383 … 0.4995458396322825 -0.33854990401453205; -0.059802030203200116 1.4806150998386711 … 0.45971417646385404 -0.1863397294559373], hyperparams = 3.0486554869187454), σimg = 2.5070588544699324, fg = 0.3539302423839184), (c = (params = [-0.8620398231896101 -0.43234643975366027 … -0.21904293949392928 0.08058409481413344; -0.5215009512292117 -1.058999213952316 … 0.38020875811599625 -1.1535429545592095; … ; 0.01765388804188612 0.4219039517669606 … 0.014745657037428818 0.8077571347264895; 0.017382042498417374 -1.1747958975301345 … -0.05181499310763299 0.24951853738604515], hyperparams = 2.898371924934943), σimg = 2.380141752482626, fg = 0.3519783085779211), (c = (params = [-0.8243939527813051 -1.0411494009977411 … 0.09616187808570598 -1.1446848194218289; -1.6055802022476977 -0.6180997958672081 … 0.08653012984667749 -0.5787422077952221; … ; 0.20160460668998564 0.2656516140557881 … -0.059235621295065384 0.07472664402820574; -0.5713603237633886 0.3884208726742153 … 0.4370295088250859 0.4372002135814307], hyperparams = 4.715566519021752), σimg = 2.0632015639325663, fg = 0.5125859190125202), (c = (params = [-0.5688438445873175 0.4588629897433608 … 0.5934060353902498 0.12461662725701868; 0.003197651793439081 -0.486637426130007 … 0.08264607059618877 -0.7290632333923197; … ; 0.41577019919934954 0.535813112955021 … 0.6603561139908398 0.9816811140209425; 0.08184522206090301 -0.2927264462904561 … -0.25541518209210556 -0.60103082707821], hyperparams = 5.338486775398733), σimg = 1.935561996229206, fg = 0.4490710650915971), (c = (params = [-0.11078147898647932 -1.1675668022911603 … -0.7290454940157883 -0.6117491625898088; -0.460178389327492 -0.41089132182802446 … 0.639028443728355 0.8256857254263639; … ; -0.33236214246538937 -0.9934716388711774 … 0.678634561115883 0.1659715848636883; -1.00970640699836 -0.021975834864290354 … 0.22005083381887028 -0.3636238640605552], hyperparams = 5.304746012940809), σimg = 2.510383423018713, fg = 0.3842967194252401), (c = (params = [0.2933745941597561 0.7205806266222277 … -0.4003298965113413 0.5808461233967768; 0.6294313703532565 -0.8843809288358137 … 0.11840009711297422 0.2757050288267131; … ; 0.662387175922236 0.5566277112522631 … 0.6421580520643807 -0.025610965981084317; -0.04639780864979078 -0.10654780489723034 … -0.07544108237955054 0.3387769592914475], hyperparams = 5.84950336593517), σimg = 2.1458911774395952, fg = 0.44268173772118347), (c = (params = [-0.013933776767618724 -0.5447082400893417 … 0.4651493146354862 -1.205167107701235; 0.2877439699642336 1.4765459806249233 … 0.3682540152497222 0.0827961951669795; … ; -0.05352861851121698 0.08945538605977275 … -0.12461092398302426 -0.10616046615713967; 0.1901797693715514 0.14691991031767987 … 0.24255794298994637 -0.8746176404271078], hyperparams = 6.072999553809627), σimg = 2.2599863040879624, fg = 0.3080731586151542), (c = (params = [0.12730952532315462 -0.4569085907216185 … 0.3588876934782149 -1.089122752925769; 0.09624718564847075 1.3434802240508072 … 0.21798098448777853 0.1255799921264312; … ; -0.13831899248334012 0.11473604614419289 … -0.03994154287339072 -0.2083270684807581; 0.20636686955492275 -0.2928803992909045 … 0.26785633414707566 -0.9887785681106557], hyperparams = 5.151098670627499), σimg = 2.24262075548013, fg = 0.30810897898968814), (c = (params = [0.29428311437288174 -0.28744919743009145 … 0.13024875494215107 -1.204623606931678; 0.4005807435139919 1.4956995747409874 … -0.1211627463034794 0.34692875000419066; … ; 0.0031370950244578233 -0.03780338757106432 … 0.08403147247593652 -0.21928475120150054; 0.11316559280399835 -0.8504239802327918 … 0.15919638258148924 -0.7800312586774694], hyperparams = 3.57920601175801), σimg = 2.1882988208290883, fg = 0.3217049077767638), (c = (params = [-0.056101170027320166 0.658627444285125 … -0.6896382534519805 0.9079926532767154; -0.1627427521273243 -0.4324886488016908 … -0.1243276071929269 -0.4992945005338458; … ; 0.24072361241022122 0.2534432177265726 … -0.4934970789607874 -0.19589466626108187; -0.1731996507106475 1.274003674845524 … -0.8889983139435934 0.4332679112447181], hyperparams = 6.031582019850623), σimg = 1.712524099391445, fg = 0.09998331865290522)], NamedTuple{(:n_steps, :is_accept, :acceptance_rate, :log_density, :hamiltonian_energy, :hamiltonian_energy_error, :max_hamiltonian_energy_error, :tree_depth, :numerical_error, :step_size, :nom_step_size, :is_adapt), Tuple{Int64, Bool, Float64, Float64, Float64, Float64, Float64, Int64, Bool, Float64, Float64, Bool}}[(n_steps = 1023, is_accept = 1, acceptance_rate = 0.9992902834360256, log_density = -471.28034446776434, hamiltonian_energy = 963.0210610573565, hamiltonian_energy_error = 0.000802326070242998, max_hamiltonian_energy_error = 0.0015327301232446189, tree_depth = 10, numerical_error = 0, step_size = 0.0001, nom_step_size = 0.0001, is_adapt = 1), (n_steps = 1023, is_accept = 1, acceptance_rate = 0.9251029951353055, log_density = -658.3531507868277, hamiltonian_energy = 996.5378095988806, hamiltonian_energy_error = 0.014666115759155218, max_hamiltonian_energy_error = -0.5345113532648611, tree_depth = 10, numerical_error = 0, step_size = 0.0015734254544819278, nom_step_size = 0.0015734254544819278, is_adapt = 1), (n_steps = 1023, is_accept = 1, acceptance_rate = 0.6137859757560344, log_density = -700.2500982979964, hamiltonian_energy = 1193.807827297034, hamiltonian_energy_error = 0.5879367981322048, max_hamiltonian_energy_error = 1.2575830648770534, tree_depth = 10, numerical_error = 0, step_size = 0.0027191098087846807, nom_step_size = 0.0027191098087846807, is_adapt = 1), (n_steps = 1023, is_accept = 1, acceptance_rate = 0.9843949456582975, log_density = -671.9100847577166, hamiltonian_energy = 1222.7838611874058, hamiltonian_energy_error = -0.010088161882777058, max_hamiltonian_energy_error = -0.08894799292966127, tree_depth = 10, numerical_error = 0, step_size = 0.0021552557635181584, nom_step_size = 0.0021552557635181584, is_adapt = 1), (n_steps = 511, is_accept = 1, acceptance_rate = 0.9917889526755399, log_density = -651.720211612358, hamiltonian_energy = 1194.8634173848823, hamiltonian_energy_error = -0.32801178340696424, max_hamiltonian_energy_error = -0.43245519085394335, tree_depth = 9, numerical_error = 0, step_size = 0.00445073451815245, nom_step_size = 0.00445073451815245, is_adapt = 1), (n_steps = 6, is_accept = 1, acceptance_rate = 0.19379703186343591, log_density = -653.8505262498163, hamiltonian_energy = 1204.430214480151, hamiltonian_energy_error = -0.1327353351259717, max_hamiltonian_energy_error = 4713.731139979011, tree_depth = 2, numerical_error = 1, step_size = 0.009765840419875316, nom_step_size = 0.009765840419875316, is_adapt = 1), (n_steps = 1023, is_accept = 1, acceptance_rate = 0.9952006433481879, log_density = -654.0012374069169, hamiltonian_energy = 1171.3378443313686, hamiltonian_energy_error = -0.004578211530997578, max_hamiltonian_energy_error = 0.01737291207950875, tree_depth = 10, numerical_error = 0, step_size = 0.0018914766041249179, nom_step_size = 0.0018914766041249179, is_adapt = 1), (n_steps = 511, is_accept = 1, acceptance_rate = 0.9374939431593299, log_density = -675.4424745670314, hamiltonian_energy = 1161.5024955125584, hamiltonian_energy_error = 0.07993217897160321, max_hamiltonian_energy_error = -0.35104153816132566, tree_depth = 9, numerical_error = 0, step_size = 0.004100680545900643, nom_step_size = 0.004100680545900643, is_adapt = 1), (n_steps = 255, is_accept = 1, acceptance_rate = 0.951548092630774, log_density = -612.865526781077, hamiltonian_energy = 1160.2813433825481, hamiltonian_energy_error = -0.31122515203151124, max_hamiltonian_energy_error = -0.9994830251566782, tree_depth = 8, numerical_error = 0, step_size = 0.007493322132379629, nom_step_size = 0.007493322132379629, is_adapt = 1), (n_steps = 3, is_accept = 1, acceptance_rate = 0.0004060093434629848, log_density = -612.865526781077, hamiltonian_energy = 1093.6591898554839, hamiltonian_energy_error = 0.0, max_hamiltonian_energy_error = 2134.8893560743313, tree_depth = 1, numerical_error = 1, step_size = 0.014299428471824607, nom_step_size = 0.014299428471824607, is_adapt = 1)  …  (n_steps = 255, is_accept = 1, acceptance_rate = 0.8128369575944936, log_density = -595.2358403071913, hamiltonian_energy = 1104.805252887043, hamiltonian_energy_error = -0.13887097842189178, max_hamiltonian_energy_error = 0.9320794108627979, tree_depth = 8, numerical_error = 0, step_size = 0.010772679071944955, nom_step_size = 0.010772679071944955, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.8579360111943513, log_density = -600.8453029539152, hamiltonian_energy = 1103.374058984637, hamiltonian_energy_error = -0.5455914256529013, max_hamiltonian_energy_error = 1.6575360337049005, tree_depth = 8, numerical_error = 0, step_size = 0.010772679071944955, nom_step_size = 0.010772679071944955, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.7561650052319309, log_density = -603.8413879111934, hamiltonian_energy = 1127.3427551292473, hamiltonian_energy_error = 0.22347481766951205, max_hamiltonian_energy_error = 1.2014543280688486, tree_depth = 8, numerical_error = 0, step_size = 0.010772679071944955, nom_step_size = 0.010772679071944955, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.9028731825693388, log_density = -644.800145080576, hamiltonian_energy = 1111.2583141445466, hamiltonian_energy_error = -0.021773877862869995, max_hamiltonian_energy_error = -0.6195969656237139, tree_depth = 8, numerical_error = 0, step_size = 0.010772679071944955, nom_step_size = 0.010772679071944955, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.890977507397043, log_density = -611.6194112069845, hamiltonian_energy = 1152.1960974881097, hamiltonian_energy_error = -0.15656378508424496, max_hamiltonian_energy_error = 2.9567809531195053, tree_depth = 8, numerical_error = 0, step_size = 0.010772679071944955, nom_step_size = 0.010772679071944955, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.9093755378732006, log_density = -632.2930145917305, hamiltonian_energy = 1126.9719432387024, hamiltonian_energy_error = -0.21379628620729818, max_hamiltonian_energy_error = -0.6191212249268574, tree_depth = 8, numerical_error = 0, step_size = 0.010772679071944955, nom_step_size = 0.010772679071944955, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.8593128880617118, log_density = -634.785078220036, hamiltonian_energy = 1155.0171384370356, hamiltonian_energy_error = 0.4666349606463882, max_hamiltonian_energy_error = 0.7416166697075823, tree_depth = 8, numerical_error = 0, step_size = 0.010772679071944955, nom_step_size = 0.010772679071944955, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.6007128239617269, log_density = -635.1073248038476, hamiltonian_energy = 1154.1266570107457, hamiltonian_energy_error = -0.7421072576626102, max_hamiltonian_energy_error = 1.6378420275475492, tree_depth = 8, numerical_error = 0, step_size = 0.010772679071944955, nom_step_size = 0.010772679071944955, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.8433004700697071, log_density = -610.8903435164816, hamiltonian_energy = 1124.8393153700308, hamiltonian_energy_error = 0.28282435304731735, max_hamiltonian_energy_error = 0.8837121971480428, tree_depth = 8, numerical_error = 0, step_size = 0.010772679071944955, nom_step_size = 0.010772679071944955, is_adapt = 0), (n_steps = 255, is_accept = 1, acceptance_rate = 0.7618050229729341, log_density = -603.2515767372013, hamiltonian_energy = 1102.7919372841984, hamiltonian_energy_error = 0.16317661893049262, max_hamiltonian_energy_error = 1.3317890751427512, tree_depth = 8, numerical_error = 0, step_size = 0.010772679071944955, nom_step_size = 0.010772679071944955, is_adapt = 0)])
Warning

This should be run for longer!

Now that we have our posterior, we can assess which parts of the image are strongly inferred by the data. This is rather unique to Comrade where more traditional imaging algorithms like CLEAN and RML are inherently unable to assess uncertainty in their reconstructions.

To explore our posterior let's first create images from a bunch of draws from the posterior

msamples = skymodel.(Ref(post), chain[501:2:end]);

The mean image is then given by

using StatsBase
imgs = intensitymap.(msamples, μas2rad(150.0), μas2rad(150.0), 128, 128)
mimg = mean(imgs)
simg = std(imgs)
fig = CM.Figure(;resolution=(400, 400));
CM.image(fig[1,1], mimg,
                   axis=(xreversed=true, aspect=1, title="Mean Image"),
                   colormap=:afmhot)
CM.image(fig[1,2], simg./(max.(mimg, 1e-5)),
                   axis=(xreversed=true, aspect=1, title="1/SNR",), colorrange=(0.0, 2.0),
                   colormap=:afmhot)
CM.image(fig[2,1], imgs[1],
                   axis=(xreversed=true, aspect=1,title="Draw 1"),
                   colormap=:afmhot)
CM.image(fig[2,2], imgs[end],
                   axis=(xreversed=true, aspect=1,title="Draw 2"),
                   colormap=:afmhot)
fig

Now let's see whether our residuals look better.

p = plot();
for s in sample(chain[501:end], 10)
    residual!(p, vlbimodel(post, s), dlcamp)
end
ylabel!("Log-Closure Amplitude Res.");
p
Example block output
p = plot();
for s in sample(chain[501:end], 10)
    residual!(p, vlbimodel(post, s), dcphase)
end
ylabel!("|Closure Phase Res.|");
p
Example block output

And viola, you have a quick and preliminary image of M87 fitting only closure products. For a publication-level version we would recommend

  1. Running the chain longer and multiple times to properly assess things like ESS and R̂ (see Geometric Modeling of EHT Data)
  2. Fitting gains. Typically gain amplitudes are good to 10-20% for the EHT not the infinite uncertainty closures implicitly assume
  3. Making sure the posterior is unimodal (hint for this example it isn't!). The EHT image posteriors can be pretty complicated, so typically you want to use a sampler that can deal with multi-modal posteriors. Check out the package Pigeons.jl for an in-development package that should easily enable this type of sampling.

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